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© Copyright 2001, Jim Loy
Part V
Go back to David Coles' Triangle Puzzle (Part IV)
Another tack: If we draw a simple odd-pointed star, and
bisect each vertex, we get a figure with only triangles in it. On the left we
see vertices from four different stars, with the vertex angle bisected. The
first is from a triangle, the second is a pentagram, the third is a heptagram
(7 vertices), and the fourth is a nonagram (9 vertices). We can deduce, from
these vertices, that the entire figures (a complete nonagram for example) will
have these quantities of lines and triangles:
figure vertices lines triangles/vertex triangles triangle 3 6 2 6 pentagram 5 10 4 20 heptagram 7 14 6 42 nonagram 9 18 8 72
The number of lines is two times the number of vertices. That is simple to show. The number of triangles per vertex increases by two as the number of vertices increases by two. I assume that this can be proven by mathematical induction (see Proof). If the pattern continues, then the next lines in the table are:
figure vertices lines triangles/vertex triangles 11-gram 11 22 10 110 13-gram 13 26 12 156 15-gram 15 30 14 210 17-gram 17 34 16 272 19-gram 19 38 18 342 21 21 42 20 420 23 23 46 22 506
22,110 and 23,114 and 24,122 and 25,130: Here is an 11-pointed star. As you can see, it continues the sequence, and breaks the record for 22 lines. We can add one line and get 23,114. Continuing as we did with 19, 20, 21, etc. lines, we get 24,122 and 25,130, all of which are records. If we continue further, we get 26,138 and 27,146. But we see that we can do better with a 13-pointed star.
26,156 etc.: Here is an 13-pointed star. As you can see, it too continues the sequence, and breaks the record for 26 lines. We can also add one line at a time, as above.
30,210 etc.: Here is an 15-pointed star. It too continues the sequence, and sets a record for 30 lines.
34,272 etc.: Here is an 17-pointed star, with 34 lines and 272 triangles.
Go to David Coles' Triangle Puzzle (Part VI).